Round-six lemma `pushLoop_cost`

Proved

NLogN.KFM.pushLoop_cost

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.KFMS9):

theorem NLogN.KFM.pushLoop_cost (Bd : RS) (dst : Fin HS) (hdO : dst ≠ mOc) (s : IState HS) (d : ℕ)
    (LD : List Item) (hD : s.G zD = .int (d : ℤ) :: LD) (v : ℕ → ℤ)
    (hB : ∀ (L : List Item) (c : ℕ), 1 ≤ c → c ≤ d →
      Bd.ok ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)) ∧
      Bd.run ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)) =
        ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)).upd dst
          (.int (v (c - 1)) :: L ++ s.G dst))
    (B C : ℕ) (hzd : zsz (d : ℤ) ≤ B) (hB1 : 1 ≤ B)
    (hbc : ∀ (L : List Item) (c : ℕ), 1 ≤ c → c ≤ d →
      Bd.cost ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)) ≤ C) :
    (RS.rep mOc (.reg zD) Bd).cost s ≤ 2 * 10 ^ 7 * (B + 2) ^ 2 +
      d * (C + 4 * 10 ^ 7 * (B + 2) ^ 2 + 30) + 32

Context. KFM S9: the cost of the source and inverse lists

Write K = 10^7 (B+2)^2. Assume every atom is at most A, with A^k < 2^B for k ≤ 8.

pmLoop_cost: the product loop costs at most (i+1)·40K.
sumLoop_cost: the sum loop costs at most (s+1)·40K.
muB_cost: one inverse costs at most (i+s+4)·40K.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Definitions.Def_IntMul_MultitapeModel
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFM
open IntMul NLogN.RS

Formal statement

theorem NLogN.KFM.pushLoop_cost (Bd : RS) (dst : Fin HS) (hdO : dst ≠ mOc) (s : IState HS) (d : ℕ)
    (LD : List Item) (hD : s.G zD = .int (d : ℤ) :: LD) (v : ℕ → ℤ)
    (hB : ∀ (L : List Item) (c : ℕ), 1 ≤ c → c ≤ d →
      Bd.ok ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)) ∧
      Bd.run ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)) =
        ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)).upd dst
          (.int (v (c - 1)) :: L ++ s.G dst))
    (B C : ℕ) (hzd : zsz (d : ℤ) ≤ B) (hB1 : 1 ≤ B)
    (hbc : ∀ (L : List Item) (c : ℕ), 1 ≤ c → c ≤ d →
      Bd.cost ((s.upd dst (L ++ s.G dst)).upd mOc (.int (c : ℤ) :: s.G mOc)) ≤ C) :
    (RS.rep mOc (.reg zD) Bd).cost s ≤ 2 * 10 ^ 7 * (B + 2) ^ 2 +
      d * (C + 4 * 10 ^ 7 * (B + 2) ^ 2 + 30) + 32 := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `NLogN.KFM.pushLoop_cost`, Solutions/KFMS9.lean (lines 296–339) of the round-six development.

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